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Completely bounded maps between sets of Banach space operators. (English) Zbl 0747.46015
Let $$X$$ and $$Y$$ be Banach spaces, let $$1\leq p<\infty$$, and let $$S$$ be a subspace of $$B(X,Y)$$, the space of all bounded operators from $$X$$ into $$Y$$. A norm on $$M_ n(S)$$, the space of all $$n\times n$$ matrices with entries in $$S$$, is obtained by $$\|(a_{ij})\|_ p=\sup(\sum_ i\|\sum_ ja_{ij}x_ j\|^ p)^{1/p}$$, the supremum being extended over all $$n$$-tuples $$(x_ j)_{j\leq n}$$ in $$X$$ such that $$\sum_ j\| x_ j\|^ p\leq 1$$.
If $$X_ 1$$, $$Y_ 1$$ are further Banach spaces, then a linear map $$u:S\to B(X_ 1,Y_ 1)$$ is called $$p$$-completely bounded $$(p.c.b)$$ if there is a constant $$C$$ such that, for any $$n$$ and any $$(a_{ij})\in M_ n(S)$$, $$\|(u(a_{ij}))\|_ p\leq C\|(a_{ij})\|_ p$$; let $$\| u\|_{pcb}$$ be the least of all such $$C$$’s. In a Hilbert space situation, the case $$p=2$$ corresponds to the usual definition of complete boundedness known from the theory of $$C^*$$-algebras.
Guided by a formal analogy of the definition of $$\|\cdot\|_{2cb}$$ with a characterization of $$L_ 2$$-factorability of Banach space operators due to J. Lindenstrauss and A. Pełczyński [Stud. Math. 29, 275–326 (1968; Zbl 0183.40501)], the author starts by giving a rather direct proof of the following generalization of G. Wittstock’s factorization theorem [J. Funct. Anal. 40, 127–150 (1981; Zbl 0495.46005)]. Let, in the above situation, $$X=Y=H$$ be a Hilbert space, and let $$u:S\to B(X_ 1,Y_ 1)$$ be 2.c.b. Then there is a Hilbert space $$\hat H$$, along with a representation $$\pi:B(H)\to B(H_ 1)$$ and operators $$V_ 1:X_ 1\to\hat H$$ and $$V_ 2:\hat H\to Y_ 1$$, such that $$u(a)=V_ 2\pi(a)V_ 1$$ for each $$a\in S$$ and $$\| V_ 1\|\cdot\| V_ 2\|\leq\| u\|_{2cb}$$, the latter inequality being then automatically an equality.
Using ultraproducts of vector valued $$L_ p$$ spaces, it is possible to extend this to the general Banach space situation $$S\subset B(X,Y)$$, $$1\leq p<\infty$$, but the statement becomes more complicated. Nevertheless, if $$X=Y_ 1=C$$, then an operator $$u$$ from $$S=Y$$ to $$B(X_ 1,Y_ 1)=X^*_ 1$$ is p.c.b. if and only it is absolutely $$p^*$$- summing $$(p^*=p/(p-1))$$, and the above extension is just Pietsch’s factorization theorem for such operators. There are numerous further special results, depending on $$X,Y$$ having particular properties, such as type, cotype,...

##### MSC:
 46B28 Spaces of operators; tensor products; approximation properties 46B08 Ultraproduct techniques in Banach space theory 46B07 Local theory of Banach spaces 46A32 Spaces of linear operators; topological tensor products; approximation properties 46L10 General theory of von Neumann algebras
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